The genus of a curve on Calabi-Yau threefolds
Abstract
Let X be a smooth projective Calabi-Yau variety of dimension $3$, with Picard number one. Let n be the self-intersection of the ample generator of the Néron-Severi group of X. Let GVg,d be the Gopakumar-Vafa invariant of X of genus g and degree d. It is conjectured that {\it if g>d2/2n+d/2+1, then GVg,d=0}. In this paper, we prove the implication when d≫ n and X⊂ ℙr is a complete intersection in projective space. The vanishing follows by proving that if C is a closed subscheme of X of dimension $1$, arithmetic genus pa(C), and degree d≫ n, then pa(C) ≤ d2/2n+d/2+1. Actually, we prove a somewhat more general result. In particular, we prove that C is an extremal curve if and only if C is the complete intersection on X with a hyperplane H and a hypersurface F, namely C=X∩ H∩ F.
Related
- The Tate Conjecture for Ordinary Calabi-Yau Threefolds
- Integral Bott--Chern cohomology of Generalized Calabi--Eckmann threefolds
- Degeneration of Calabi-Yau threefolds with Galois action induced by modular form of weight 3
- Stability conditions on Calabi-Yau threefolds of type K
- Rigid and obstructed tangent bundles on Calabi--Yau threefolds
BibTeX
@misc{gennaro2026genus,
title = {{The genus of a curve on Calabi-Yau threefolds}},
author = {Vincenzo Di Gennaro and Antonio Rapagnetta and Pietro Sabatino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25065},
eprint = {2610.25065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25065},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The genus of a curve on Calabi-Yau threefolds
AU - Vincenzo Di Gennaro
AU - Antonio Rapagnetta
AU - Pietro Sabatino
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25065
UR - https://arxiv.org/abs/2610.25065
AB - Let X be a smooth projective Calabi-Yau variety of dimension $3$, with Picard number one. Let n be the self-intersection of the ample generator of the Néron-Severi group of X. Let GV_g,d be the Gopakumar-Vafa invariant of X of genus g and degree d. It is conjectured that {\it if g>d^2/2n+d/2+1, then GV_g,d=0}. In this paper, we prove the implication when d≫ n and X⊂ ℙ^r is a complete intersection in projective space. The vanishing follows by proving that if C is a closed subscheme of X of dimension $1$, arithmetic genus p_a(C), and degree d≫ n, then p_a(C) ≤ d^2/2n+d/2+1. Actually, we prove a somewhat more general result. In particular, we prove that C is an extremal curve if and only if C is the complete intersection on X with a hyperplane H and a hypersurface F, namely C=X∩ H∩ F.
ER -